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Nov. 2014, Volume 8, No. 11 (Serial No. 84), pp. 1445-1452 Journal of Civil Engineering and Architecture, ISSN 1934-7359, USA
Dynamic Characteristic of the Medium Span Concrete
Footbridges
Marek Pańtak
Department of Bridges and Tunnels Construction, Cracow University of Technology, Cracow 31-155, Poland
Abstract: Concrete footbridges, due to their mass, stiffness and damping, are perceived as structures more resistant to vibration caused by dynamic action of the users. In order to verify the dynamic behaviour of concrete footbridges, a series of field tests and numerical analyses have been carried out. In the paper, the results of the dynamic field tests of three medium span concrete footbridges with different structural systems (frame, beam and arch footbridges) and their dynamic characteristics (mass, stiffness and damping) are presented. The field tests were carried out for different types of vibration excitation caused by walking, running and jumping persons. Furthermore, the vibrational comfort criteria for footbridges are shortly described and verified for examined structures. The study were supplemented by numerical calculation of natural mode shapes and frequencies of the structures using the 3D FEA (finite element analysis) models with elastic supports elements in order to ensure the compatibility of the calculated and measured mode shapes of the footbridges.
Key words: Footbridge vibrations, concrete footbridges, comfort criteria.
1. Introduction
Properly designed civil engineering structures
should fulfil the requirements of two principal limit
states: the ultimate and the serviceability limit state.
The conditions of the serviceability limit state require
among other things to limit the structure vibrations.
In the case of footbridges, acceptable levels of
vibration acceleration, resulting from the requirements
of the vibrational comfort criteria for walking people,
are in the range of av,max = 0.5-1.0 m/s2 for vertical
vibrations and ah,max = 0.1-0.2 m/s2 for horizontal
vibrations [1-7]. For rare and special dynamic loads
(impact of large crowd or intentional excitation (act of
vandalism)), acceptable levels of vibration
acceleration are in the range of avr,max = 0.85-1.2 m/s2
for vertical vibrations and ahr,max = 0.2-0.3 m/s2 for
horizontal vibrations [1-7]. If these vibration comfort
requirements are exceeded more than once a week, it
is advisable to take a countermeasures to change the
Corresponding author: Marek Pańtak, Dr., research fields:
bridge design, construction and bridge dynamics. E-mail: mpantak@pk.edu.pl.
dynamic characteristics of the structure by modifying
the structural system or by installing vibration
dampers.
Generally, the concrete structures are characterised
by high mass, high stiffness and in some cases also by
high damping. These main features of the concrete
structures can lead to their low dynamic susceptibility
and low vibrations amplitudes. The concrete
footbridges are perceived as structures more resistant
to vibration caused by dynamic action of users than
steel or composite footbridges. In further part of the
article, the dynamic characteristic of three concrete
footbridges obtained during numerical dynamic
analyses and the field tests of a beam, rigid frame and
arch footbridges (Fig. 1) with medium span length
(35.0-70.0 m) are presented.
2. Dynamic Characteristic of Tested Concrete Footbridges
2.1 Footbridge over A4 Motorway between
Stanisławice/Kłaj Rest Area, Poland
The footbridge was built in July, 2011. This is a
DAVID PUBLISHING
D
Dynamic Characteristic of the Medium Span Concrete Footbridges
1446
three-span frame footbridge with spans 14.0 m +
34.0 m + 14.0 m and with total length of the main
structures 64.0 m. Access to the footbridge is ensured
by stairs and ramps for disabled persons (Fig. 1a). The
structural system is a rigid frame with inclined piers
rigidly connected with girders and with foundation
plates. Both foundation plates (6.0 m × 5.0 m × 1.2 m)
are founded on 18 inclined concrete piles Ø500 mm.
Main girder is vertically curved (the radius of
curvature R = 330.0 m) and is made with prestressed
concrete C45/55. The inclined footbridge piers (0.5 m
thick) are made with reinforced concrete C35/45. The
width of the piers varies from 1.5 m at the connection
with the foundation plate to 1.0 m at the connection
with the main girder. The footbridge deck is
connected with ramps for disabled persons by means
of composite steel-rubber expansion joints cover. The
width of the dilatation gap on the both ends of the
footbridge is 30 mm. In Fig. 2, the longitudinal
section and cross section of the footbridge are
presented.
The dynamic field tests of the footbridge were
carried out in September, 2013. In numerical dynamic
analysis of natural frequency and mode shapes of the
footbridge, the time dependent material constants,
after two years of footbridge operation (t = 730 d),
were used according to Ref. [8]: for footbridge deck
C45/55: Ecm(t) = 40.0 GPa, γm = 2,600 kg/m3—density
of concrete; for footbridge piers C35/45: Ecm(t) = 37.0
GPa, γm = 2,500 kg/m3. By accepting a higher density
of concrete for the footbridge deck in computational
FEA (finite element analysis) model, the additional
weight of railings, plexiglass screens and steel pillars
on the footbridge deck (170 kg/m) have been taken
into account in calculations of natural frequencies and
mode shapes of the footbridge.
(a)
(b)
(c)
Fig. 1 Tested footbridges: (a) footbridge over A4 motorway within Stanisławice/Kłaj Rest Area, Poland; (b) footbridge over Powstanców Śląskich Av. in Kraków, Poland; (c) footbridge over D5 motorway near Cekov, Czech Republic [9].
2% 2%
0.15
m
1.00 m 1.55 m 1.55 m
0.55 m 0.55 m
4.10 m
3.00 m
2.40
m
0.30
m
0.60
m
P lexiglas sheet (screens)
Epoxy resin pavement 5.0 mm Concrete girder C60/75
Fig. 2 Footbridge over A4 motorway between Stanisławice and Kłaj Rest Area.
Dynamic Characteristic of the Medium Span Concrete Footbridges
1447
The computational FEA model of the footbridge
was prepared in AASP (Autodesk Algor Simulation
Professional). The 3D model was built using 3D brick
elements available in the AASP software. Because the
bridge is a rigid frame structure for further analysis, it
was important to properly model the connection
between the piers and the foundation plates. Due to
the lack of detailed data of soil parameters, the
stiffnesses of spring (elastic) connection in X, Y and Z
direction (kx, ky, kz) for translational and rotational
degrees of freedom of the footbridge foundations were
assumed to ensure the compatibility of the first
calculated and measured mode shapes of the
footbridge. The nodal boudary conditions for
footbridge piers were modeled as spring supports with
stiffness of the spring for translational DOF (degree of
freedom) kTx = 10.80 GN/m, kTy = kTz = 3.60 GN/m
for each footbridge pier. The stiffnesses of the spring
elements for rotational DOF kRx = kRy = kRz had no
impact on the footbridge natural frequency and were
modeled as fully constrained (X—longitudinal axis of
the footbridge, Y—transverse axis, Z—vertical axis).
Connection between footbridge deck and the ramps
for disabled persons due to incorrect implementation
of composite steel-rubber expansion joints cover (in a
very small dilatation gap) was modeled with
constrained X translation. For Y and Z translations,
kTy = kTz = 1.70 MN/m was assumed. The rotational
DOF for Y and Z rotations were modeled as unblocked
(free) due to a possibility of deformation of the rubber
expansion joint cover in these directions.
During the numerical modal analysis, several mode
shapes with frequency in the frequency range of
human dynamic impact during walking, running or
jumping were calculated (Fig. 3). During the field
tests, the first two mode shapes were identified with
frequency ft1 = 1.14 Hz and ft2 = 2.35 Hz. The
dynamic tests were performed by walking, running
and jumping persons. The vibration acceleration was
measured by three two-axis accelerometers mounted
on the edge of the footbridge deck in 1/2 L, 1/3 L and
2/3 L (L—length of the footbridge span). To identify
the mode shapes of the footbridge, the correlation of
the measuring signal from each accelerometers in two
directions (vertical and horizontal transverse to
footbridge axis) was verified. The footbridge
vibrations were induced by one to three walking,
running or jumping persons using the metronome set
at a rate corresponding to the natural vibration
frequencies of the footbridge.
The first mode shape of the footbridge was very
often excited by vans, lorries and buses passing under
the footbridge. The amplitude of the horizontal
acceleration was very small a1h = 0.04 m/s2 and did
not impair the comfort of use of the footbridge
(acceptable value ah, max = 0.1-0.2 m/s2).
The second mode shape can be easily excited by
slowly running persons and by jumping persons
(intentional excitation and act of vandalism). The
amplitudes of the vibration acceleration reached the
values of a1v = 0.33 m/s2 (during excitation by one
running person), a2v = 0.42 m/s2 (during excitation by
one jumping person) and a3v = 0.81 m/s2 (during
excitation by three jumping persons).
According to the guidelines presented in Ref. [3],
the footbridge can be classified into Class IV (seldom
used footbridge, built to link sparsely populated areas
or to ensure continuity of the pedestrian footpath in
(a) (b) (c) (d)
Fig. 3 The first four mode shapes of the footbridge obtained during numerical modal analysis: (a) horizontal (transversal) f1 = 1.16 Hz; (b) vertical f2 = 2.33 Hz; (c) horizontal (transversal) f3 = 2.57 Hz; (d) vertical f4 = 5.13 Hz.
Dynamic Characteristic of the Medium Span Concrete Footbridges
1448
motorway or express lane areas). Class IV footbridges
are considered not to require any calculation to check
dynamic behaviour. For very light footbridges, it
seems advisable to select at least Class III (footbridge
for standard use, which may occasionally be crossed
by large groups of people but that will never be loaded
throughout its usable area) to ensure a minimum
amount of vibration control.
According to the criteria presented in Refs. [2, 3, 7],
the footbridge can be placed in the group of structures
rarely excited by pedestrians. In this case, the
acceptable levels of vibration acceleration are in the
range of avr, max = 0.85-1.2 m/s2 for vertical vibrations
and ahr, max = 0.2-0.3 m/s2 for horizontal vibrations.
The requirements of the comfort criteria are fulfilled
in case of all rare occurring conditions of use such as
running or jumping. Under the normal conditions of
use, the comfort requirements are met with a wide
margin of safety (with very small risk of exceeding).
The deck vibrations are not felt by walking persons
but are felt by standing persons what was confirmed
during the tests.
In further analysis, the structural stiffness, mass of
the superstructure and logarithmic decrement were
calculated (Table 1).
2.2 Footbridge over Powstanców Śląskich Av. in
Kraków, Poland
The footbridge was built in 2005. This is a
two-span girder footbridge with two girders of
variable height, vertically curved deck (the radius of
curvature R = 980.0 m) and spans of 40.0 + 40.0 m.
The height of the girders varies from 1.3 m to 2.1 m.
The usable width of the footbridge is 4.0 m. The
girders are made with prestressed concrete C30/37
(Ecm = 32.0 GPa, γm = 2,600 kg/m3) and are supported
on three concrete piers by means of elastomeric
bearings. On the central pier, the girders are
additionally supported by means of two steel hot
rolled beams HEA100 mounted on both sides of the
elastomeric bearings beside the bearings concrete pads
(bearings seats). The steel beams were mounted after
the cracking of the concrete girders over the central
pier during the footbridge operation to limit the
rotation of the girders around the horizontal axis
transverse to the axis of the footbridge. In Fig. 4, the
longitudinal section and cross section of the
footbridge are presented.
Table 1 Structural stiffness, logarithmic decrement and the mass of the superstructure of the footbridge.
Frequency (Hz)
Structural stiffness (kN/mm) Logarithmic decrement δ Mass of the superstructure (deck with screens and railings) (t/m)
2.35 14.75 0.052-0.058 4.50
4.00 m
4.60 m
1.30 m
2.1
0 m
0.30 m 0.30 m
0.40 m 0.40 m 1.75 m 1.02 m 1.02 m
1.22 m 2.15 m 1.22 m
0.30 m
0.3
0 m
2% 2% 0.20
m
Epoxy resin pavement 5.0 mm
Concrete girder C30/37
Fig. 4 Footbridge over Powstanców Śląskich Av. in Kraków, Poland.
Dynamic Characteristic of the Medium Span Concrete Footbridges
1449
The computational FEA model of the footbridge
was prepared in AASP. The 3D model was built using
2D plate elements available in the AASP. The
supports of the girders on the central pier were
modeled with constrained X, Y and Z translation (fixed
bearing) and constrained X, Y and Z rotation due to
limited rotation of the girders supported on the central
pier by means of bearings and additional steel beams
(X—longitudinal axis of the footbridge, Y—transverse
axis, Z—vertical axis). The kinematic constraints were
placed in nodes of FEA model in place and over a
length corresponding to the accurate location of the
support points in a real structure. The supports on the
both ends of the footbridge were modeled with
constrained Y and Z translation, elastic X translation
(kx = 800.0 MN/m, assumed taking into account
elasticity of elastomeric bearings and rubber
expansion joints cover) and constrained X rotation.
The Y and Z rotations were modeled as unblocked
(free).
During the numerical modal analysis, the modal
shapes presented below were calculated (Fig. 5).
The dynamic field tests were performed with
frequency ft1 = 2.44 Hz, identified during the tests,
and often induced during everyday use of the
footbridge as observed during the monitoring of the
footbridge. The footbridge vibrations were induced by
one to three walking, running and jumping persons.
Three two-axis accelerometers were mounted on the
edge of the footbridge deck in 1/2 L on both spans and
in 1/3 L on the left span. To identify the mode shapes
of the footbridge, the correlation of the measuring
signal was verified. The resonant vibration of the
footbridge was induced using the metronome set at a
rate corresponding to the natural vibration frequencies
of the footbridge (146 beats/min.).
The recorded maximal vibration accelerations were:
a1v = 0.15 m/s2 (during excitation by one running
person), a2v = 0.27 m/s2 (during excitation by three
running persons), a3v = 0.33 m/s2 (during excitation by
one jumping person), a4v = 0.43 m/s2 (during
excitation by three jumping persons). The footbridge
can be classified into Class III and the acceptable level
of vibration acceleration can be assumed as for rare
occurring vertical vibration avr,max = 0.85-1.2 m/s2.
However, the footbridge fulfils the requirements of
the vibrational comfort criteria both for rare and
everyday occurring vibrations. The comfort of use of
the footbridge will not be impaired but the vibrations
of the footbridge deck will be clearly felt by standing
persons in case of their excitation by runners or
vandals. The other characteristics of the footbridge are
presented in Table 2.
2.3 Footbridge over D5 Motorway near Cekov, Czech
Republic
The footbridge is a single span arch footbridge with
reinforced concrete deck connected with reinforced
concrete arch by means of steel square pipes (200 ×
200 × 20 mm) filed with polyurethane foam. The arch
is supported on concrete inclined foundation plates by
means of steel bearings. The span of the arch is
69.00 m and the rise of the arch is 3.34 m. Usable
width of the deck is 3.00 m and its longitudinal slope
(a) (b) (c) (d)
Fig. 5 The first four mode shapes of the footbridge obtained during numerical modal analysis: (a) vertical f1 = 2.45 Hz; (b) vertical f2 = 2.69 Hz; (c) torsional f3 = 3.88 Hz; (d) torsional f4 = 4.47 Hz.
Table 2 Structural stiffness, logarithmic decrement and the mass of the superstructure of the footbridge.
Frequency (Hz)
Structural stiffness (kN/mm) Logarithmic decrement δ Mass of the superstructure (girders with deck and railings) (t/m)
2.44 10.45 0.048-0.054 5.90
Dynamic Characteristic of the Medium Span Concrete Footbridges
1450
5.5%. The length of steel pipes connecting the arch
and the deck varies from 5.40 m to 0.275 m. The
footbridge was built in 1995.
The dynamic field tests of the footbridge were
carried out in August, 2010. In numerical dynamic
analysis of natural frequency and mode shapes of the
footbridge, the time dependent material constants,
after 15 years of footbridge operation (t = 5,400 d),
were used according to Ref. [8]. The deck was
modeled as a concrete plate with variable thickness
0.35-0.50 m made of concrete C35/45 (Fig. 6), (Ecm(t)
= 38.0 MPa, γm = 2,600 kg/m3—equivalent density of
the deck with asphalt pavement and railings). The arch
was modeled as a concrete plate with thickness 0.80 m
made of concrete C45/55, (Ecm(t) = 40.0 MPa, γm =
2,500 kg/m3). In Fig. 4, the longitudinal section and
cross section of the footbridge are presented.
The computational FEA model of the footbridge
was prepared in AASP. The 3D model was built using
2D plate elements and 2D beam elements available in
the AASP. The supports of the arch were modeled
with constrained X, Y and Z translations, constrained X,
Z rotations and elastic Y rotation (kRy = 40.0
MNm/deg). The supports of the deck on both ends
were modeled with elastic X translation (kx = 600.0
MN/m) constrained Y and Z translations, constrained
X, Y rotation and unconstrained (free) Z rotation.
Fundamental mode shapes of the footbridge are
presented in Fig. 7. During the field tests, the first
three mode shapes were identified with frequencies: ft1
= 1.37 Hz, ft2 = 2.12 Hz and ft3 = 2.49 Hz.
The vibration excitations during the field tests were
performed for frequency ft3 = 2.49 Hz. The footbridge
vibrations were induced by one to three walking,
running and jumping persons. The two-axis
accelerometers were mounted on the edge of the
footbridge deck in 1/2 L and in 1/3 L. To identify the
mode shapes of the footbridge, the correlation of the
measuring signal was verified. The resonant vibration
of the footbridge was induced using the metronome
set at a rate corresponding to the natural vibration
frequencies of the footbridge (150 beats/min.).
The maximal amplitudes of the vibration
acceleration recorded during the tests reached the
values of a1v = 0.20 m/s2 (during excitation by one
running person), a2v = 0.48 m/s2 (during excitation by
three running persons), a3v = 0.52 m/s2 (during
excitation by three jumping persons). The requirements
2.00 m
2% 2%
3.40 m 0.20 m 0.20 m 3.00 m
0.70 m 0.70 m
1.20
m
0.35
m
0.80
m
3.40 m 0.20 m 0.20 m 3.00 m
2% 2% 1.
20 m
0.
50 m
3.
00 m
0.
80 m
2.00 m 1.40 m 0.30 m 0.30 m
0.10
m
200 x 200 x 20 mm Square steel pipe
Fig. 6 Footbridge over D5 motorway near Cekov, Czech Republic.
(a) (b) (c) (d)
Fig. 7 The first four mode shapes of the footbridge obtained during numerical modal analysis: (a) horizontal (transversal) f1 = 1.46 Hz; (b) vertical f2 = 2.15 Hz; (c) vertical f3 = 2.36 Hz; (d) vertical f4 = 3.55 Hz.
Dynamic Characteristic of the Medium Span Concrete Footbridges
1451
Table 3 Structural stiffness, logarithmic decrement and the mass of the superstructure of the footbridge.
Frequency (Hz)
Structural stiffness (kN/mm) Logarithmic decrement δ Mass of the superstructure (arch with deck and railings) (t/m)
2.49 24.10 0.098-0.110 5.45
of the comfort criteria on the footbridge are fulfilled
with a wide margin of safety also in case of
intentional vibration excitation. The footbridge is
located near Cekov village in the agricultural region
and, according to Ref. [3], can be classified into Class
IV. The footbridges classified in Class IV do not
require dynamic analysis and verification of the
comfort of use criteria. However, this is an example of
very stiff and good designed footbridge which ensures
the comfort of use even during an intensive use
defined in Ref. [3] for footbridges in Class I (urban
footbridges linking up high pedestrian density areas,
frequently used by dense crowds, subjected to very
heavy traffic).
In Table 3, the structural stiffness, mass of the
superstructure and logarithmic decrement of the
footbridge are presented.
3. Discussion and Results
The results of dynamic field tests show that natural
frequencies of all tested concrete footbridges are in a
range of frequency of the users’ vertical dynamic
impact during fast walking, running and jumping
(2.20-3.00 Hz) [10]. In this cases, it is possible to
induce the resonant vibration of the structure during
its normal use, especially by slow running persons
(2.20-2.70 Hz—frequency range of vertical dynamic
action during slow running). In case of the first rigid
frame footbridge over A4 motorway, it is also
possible to induce the resonant vibration of the first
horizontal mode shapes by walking users (0.80-1.20
Hz—frequency range of horizontal dynamic action
during walking).
The vibration accelerations recorded during the
dynamic tests on examined footbridges were in the
range of av = 0.15-0.80 m/s2, ah = 0.02-0.04 m/s2 and
did not exceed the limit values for vertical vibrations
av,max = 0.5-1.0 m/s2 and for horizontal vibrations
ah,max = 0.1-0.2 m/s2. Further analyses show that
obtained results depend mainly on the high stiffness
and mass of the investigated footbridges. The
logarithmic decrement of damping δ for tested
footbridges is in the range of low damping (δ < 0.10).
As set forth in Ref. [11] on the basis of the dynamic
field tests of 35 footbridges, 50% of tested footbridges
have δ = 0.05 ~ 0.10, 30% of which have δ = 0.03 ~
0.05). Only 11% footbridges had δ = 0.10 ~ 0.15.
Characteristic for the footbridges is also lack of the
structures with a large damping (δ > 0.20).
In case of steel footbridges, the logarithmic
decrement of damping δ is very often in the range of
0.01-0.02 [7, 12]. The stiffness of the medium span
steel footbridges is in the range of 1.5-3.0 kN/mm and
their mass per unit length is 0.10-0.50 t/m. It leads to
general conclusion that investigated concrete
footbridges mostly due to their large stiffness
(10.0-25.0 kN/mm) and large mass (4.0-6.0 t/m) are
much more resistant to dynamic impact of the
pedestrians than lightweight and flexible steel
footbridges. The further researches concerning the
determination of influence of structural mass and
stiffness on the dynamic response of the footbridges
are recommended.
During the numerical calculation of natural
frequencies of the footbridges, the high impact of the
supports elasticity/stiffness and elasticity/stiffness of
expansion joint cover into obtained results was
observed. In the computational FEA models, the
elastic (spring) supports matched by successive
attempts were used to ensure the compatibility of the
calculated and measured mode shapes of the
footbridges. It is recommended to elaborate a
recommendation and more accurate method of
estimation of the stiffnesses of different types of
bridge bearings and stiffnesses of the expansion join
cover for their use in dynamic analyzes.
Dynamic Characteristic of the Medium Span Concrete Footbridges
1452
4. Conclusions
In all presented examples of the medium span
concrete footbridges, the requirements of vibrational
comfort criteria were fulfilled with a wide margin of
safety for the most probable cases of dynamic loads
during footbridge exploitation. The logarithmic
decrement of the tested concrete footbridges, although
it is relatively higher than in steel footbridges, is still
in the range of low damping as in the steel and
composite footbridges. However, in comparison with
steel and composite footbridges, analysed concrete
footbridges are characterised by higher structural
stiffness and higher mass of superstructure. These
features of the concrete structures ensure the
fulfilment of the comfort criteria requirements also for
higher spans length than in case of lightweight steel or
composite structures.
References
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[2] M. Pańtak, A. Flaga, Experimental verification of own vibration comfort criteria for footbridges, in: Proceedings of the Seminar Wroclaw Bridge Days “Footbridges, Architecture, Design, Construction, Research”, Educational Publishing House of Lower Silesia, Wrocław, 2007, pp. 233-246. (in Polish)
[3] Footbridges—Assessment of Vibrational Behaviour of Footbridges Under Pedestrian Loading, Service d'études techniques des routes et autoroutes—Sétra (Road and
Highway Technical Studies Department), Sétra, French, 2006.
[4] H. Bachmann, W. Ammann, F. Deischl, J. Eeisenmann, I.
Floegl, G.H. Hirsch, et al., Vibration Problems in
Structures—Practical Guidelines, Birkhäuser Verlag,
Germany, 1995.
[5] H. Bachmann, W. Ammann, Vibrations in Structures
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[6] EN 1990:2002/A1:2005, Eurocode: Basis of Structural
Design: Annex A2, CEN (European Committee for
Standardization), 2004.
[7] M. Pańtak, B. Jarek, W. Średniawa, Application of EN
1990/A1 vibration serviceability limit state requirements
for steel footbridges, Procedia Engineering 40 (2012)
345-350.
[8] EN 1992-1-1:2004, Eurocode 2: Design of Concrete
Structures. Part 1-1: General Rules and Rules for
Buildings, CEN (European Committee for
Standardization), 2004.
[9] R. Menšík, V. Kanický, V. Salajka, Footbridge over D5
near Cekov, DOSING (Dopravoprojekt Brno Group Ltd.)
[Online], http://www.dosing.cz (accessed Dec. 4, 2013).
[10] H. Bachmann, “Lively” footbridges—A real challenge, in:
Proceedings of the International Conference “Footbridge
2002”, Paris, 2002, pp. 18-30.
[11] M. Salamak, The role of damping in footbridges and
methods of its identification, in: Proceedings of the
Seminars “Design, Construction and Aesthetics of
Footbridges”, Cracow, 2003, pp. 189-213. (in Polish)
[12] M. Pańtak, Dynamic characteristics of medium span truss,
cable-stayed and suspension steel footbridges under
human-induced excitation, in: Proceedings of the
International Conference “Footbridge 2011 Attractive
Structures at Reasonable Costs”, Wrocław, 2011, pp.
1209-1214.
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